On almost commutative Friedmann–Lemaître–Robertson–Walker geometries

Abstract

We analyze the leading terms of the spectral action for a model of noncommutative geometry, which is a product of 4-dimensional Riemannian manifold with a two-point space exploring the previously neglected case when the metrics over each sheet are different. Assuming the Friedmann-Lemaître-Robertson-Walker type of the metric for both sheets we obtain the action, which in addition to the the usual cosmological constant terms and the Einstein-Hilbert term involves a nonlinear interaction term. We study qualitative picture of potential consequences of such term in the basic cosmological models.

On stability of Friedmann-Lemaître-Robertson-Walker solutions in doubled geometries

“…Our conclusion is that for the considered range of models, including flat and curved spatial geometries with dark-energy, radiation or matter dominance there exist a range of parameters so that the symmetric solution (product geometry) is dynamically stable. Our analysis confirms but hugely extends the earlier indications [12] by allowing both the scale factors as well as lapse functions to vary. The stability of the cosmological solutions suggests that the models with two metrics are admissible from the physical point of view and are an interesting modification of geometry that may be used in future models.…”

“…Although the topology of the flat case in physics is not exactly toroidal, from the point of view of local behaviour it is identical to such, which was already analysed for b = 1 in 5 [12]. In this section we generalize those results to the case with arbitrary function b(t), so we consider here toroidal Friedmann-Lemaître-Robertson-Walker geometries described by the following metric in the coordinate system (t, x) = (t, x 1 , x 2 , x 3 ):…”

“…Furthermore, the action functional was derived for the Euclidean model and to pass to physical situation we need to perform Wick rotation, as described in the [12]. In our case, this will affect only the square of the time derivative of the scaling factors a i (t), which will change signs.…”

“…In particular, the first question posed is whether the two metrics interact with each other. A first step in this direction was done in [12], where a simple model of two Friedmann-Lemaître-Robertson-Walker-type, flat geometries with identical lapse function was considered, resulting in the effective potential term linking the two geometries.…”

Spectral interaction between universes

“…In the constructions so far, one usually assumed the natural product-type geometry (producttype Dirac operators), which, after applying the spectral action procedure [7,8] led to the standard Einstein-Hilbert action for the metric, identical on the two universes. Yet this is not the most general form of the Dirac operator and different metrics on the two separate universes are admissible [9]. Together with the Higgs-type field that mediates between the two geometries one can obtain an interaction term between the two metrics, leading to an interesting class of models, which appear to be viable from the point of view of cosmological models [10].…”